A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints

Let K n be the cone of positive semidefinite n X n matrices and let Å be an affine subspace of the space of symmetric matrices such that the intersection K n ∩Å is nonempty and bounded. Suppose that n ≥ 3 and that codim Å = r+2 choose 2 for some 1 ≤ r ≤ n-2 . Then there is a matrix X ∈ K n ∩Å such that rank X ≤ r . We give a short geometric proof of this result, use it to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and describe its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42424/1/454-25-1-23_10074.pd

Polynomial diffusions on compact quadric sets

Polynomial processes are defined by the property that conditional expectations of polynomial functions of the process are again polynomials of the same or lower degree. Many fundamental stochastic processes, including affine processes, are polynomial, and their tractable structure makes them important in applications. In this paper we study polynomial diffusions whose state space is a compact quadric set. Necessary and sufficient conditions for existence, uniqueness, and boundary attainment are given. The existence of a convenient parameterization of the generator is shown to be closely related to the classical problem of expressing nonnegative polynomials---specifically, biquadratic forms vanishing on the diagonal---as a sum of squares. We prove that in dimension $d\le 4$ every such biquadratic form is a sum of squares, while for $d\ge6$ there are counterexamples. The case $d=5$ remains open. An equivalent probabilistic description of the sum of squares property is provided, and we show how it can be used to obtain results on pathwise uniqueness and existence of smooth densities.Comment: Forthcoming in Stochastic Processes and their Application

Low-Rank Univariate Sum of Squares Has No Spurious Local Minima

We study the problem of decomposing a polynomial $p$ into a sum of $r$ squares by minimizing a quadratically penalized objective $f_p(\mathbf{u}) = \left\lVert \sum_{i=1}^r u_i^2 - p\right\lVert^2$. This objective is nonconvex and is equivalent to the rank-$r$ Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials $p$, if $r \ge 2$ then $f_p(\mathbf{u})$ has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, $r$ has to be roughly the square root of the number of constraints (the degree of $p$) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally-spaced points on the circle, the gradient $\nabla f_p$ can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials.Comment: 18 pages, to appear in SIAM Journal on Optimizatio